Slaviša Šalinić

On the brachistochrone of a variable mass particle in general force fields

Abstract The problem of the brachistochronic motion of a variable mass particle is considered. The particle moves through a resistant medium in the field of arbitrary active forces. Beginning from these general assumptions, and applying Pontryagin’s minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained and solved. The solution proposed involves an appropriate numerical procedure based upon the shooting method. In this numerical procedure, the evaluation of ranges for unknown values of costate variables is avoided by the choice of a corresponding Cartesian coordinate of the particle as an independent variable. A numerical example assuming the resistance force proportional to the square of the particle speed is presented. A review of existing results for related problems is provided, and it can be shown that these problems may be regarded as special cases of the brachistochrone problem formulated and solved in this paper under very general assumptions by means of optimal control theory.

On the brachistochronic motion of a variable-mass mechanical system in general force fields

The problem of the brachistochronic motion of a mechanical system composed of rigid bodies and variable-mass particles is solved. The laws of the time-rate of mass variation of the particles as well as relative velocities of the expelled (or gained) masses are assumed to be known. The system moves in an arbitrary field of known potential and nonpotential forces. Applying Pontryagin’s minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained. The appropriate numerical procedure based on the shooting method to solve the obtained two-point boundary value problem is presented. The considerations in the paper are illustrated by an example of determining the brachistochronic motion of a system composed of a rigid rod and two variable-mass particles attached to the rod.

A rigid multibody method for free vibration analysis of beams with variable axial parameters

This paper presents a new approach to the problem of determining the frequencies and mode shapes of Euler–Bernoulli tapered cantilever beams with a tip mass and a spring at the free end. The approach is based on the replacement of the flexible beam by a rigid multibody system. Beams with constant thickness and exponentially and linearly tapered width, as well as double-tapered cantilever beams are considered. The influence of the tip mass, stiffness of the spring, and taper on the frequencies of the free transverse vibrations of tapered cantilever beams are examined. Numerical examples with results confirming the convergence and accuracy of the approach are given.

A novel approach to an automated needle insertion in brachytherapy procedures

One of the most challenging phases in interstitial brachytherapy is the placement of the needles. In these medical procedures, the needles are inserted inside the tissue to guide the positioning of the radioactive sources. The low-dose-rate radioactive sources are placed inside the tissue permanently, whereas a radioactive source in the high-dose-rate brachytherapy is temporarily placed in the desired positions so that the delivery of the prescription dose to the clinical targets can be achieved. Consequently, the precise needle placement directly influences the radiation dose delivery and the treatment outcomes of patients. Any deviation from the desired position of the radioactive sources can cause a suboptimal dose distribution and inadequate tumor coverage. Therefore, it is of significant importance to develop a robust and sophisticated tool that can perform the automatic needle placement with a high level of accuracy for different medical procedures and conditions. In this study, we propose a novel concept for the automatic needle insertion using a new miniature automated robotic system. The mathematical model of this system was presented in detail, allowing the implementation of the model predictive control that can be used to govern the mechanism. The purpose of this approach was to minimize the lateral components of the generalized reactive force which is responsible for the tissue displacement and, consequently, for the needle deflection. The proposed approach was designed to predict and to compensate for the unmeasured disturbances, such as needle deflection or tissue resistance and reactive force, and it was capable of correcting them without waiting until the effect appears at the output of the system causing the needle deviation from the desired positions. The extensive simulation of the system was presented to evaluate the feasibility of the method and the parameters of interest including displacements, system errors and system responses to the change in the environmental conditions.

The brachistochronic motion of a vertical disk rolling on a horizontal plane without slip

This paper deals with the brachistochronic motion of a thin uniform disk rolling on a horizontal plane without slip. The problem is formulated and solved within the frame of the optimal control theory. The brachistochronic motion of the disk is controlled by three torques. The possibility of the realization of the brachistochronic motion found in presence of Coulomb dry friction forces is inspected. Also, the influence of values of the coefficient of dry friction on the structure of the extremal trajectory is analyzed. Two illustrative numerical examples are provided.

Brachistochronic motion of a nonholonomic variable-mass mechanical system in general force fields:

In this paper, the brachistochronic motion of a mechanical system composed of variable-mass particles is analysed. Workless (ideal) holonomic and linear nonholonomic constraints are imposed on the system. It is assumed that the system moves in an arbitrary field of known potential and nonpotential forces with prescribed both laws of the time-rate of mass variation of the particles and relative velocities of the expelled (or gained) masses. The first time-derivatives of quasi-velocities are taken as control variables. Using Pontryagin’s maximum principle and singular optimal control theory, the problem of brachistochronic motion of the nonholonomic variable-mass mechanical system is solved as a two-point boundary value problem. In addition, a discussion about the realization of control forces is given. The results are illustrated via an example.

A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

A new approach for the determination of the global minimum time for the case of the brachistochronic motion of the Chaplygin sleigh is presented. The new approach is based on the use of the shooting method in solving the corresponding two-point boundary-value problem and defining either the crossing points of surfaces or the crossing points space of curves in a three-dimensional space of two costate variables and the time of the brachistochronic motion of the sleigh. A number of examples for multiple extremals of the Chaplygin sleigh brachistochrone problem are provided. In these examples, the global minimum is the solution to which the minimum time of motion corresponds.

A mathematical model of a novel automated medical device for needle insertions

In medical procedures, needle insertion is a challenging task highly dependent on the surgeons manual skills. Implanted needles are used for drug delivery, biopsy, delivery of radiation sources, etc. In the named clinical procedures, the accuracy of the needle placement is crucial for patient treatment outcomes. Therefore, we have proposed an automated medical device for needle implantation to eliminate uncertainties of the standard procedures and to increase the accuracy of the needle placement. In this article, the mathematical model of such a device has been developed. The mathematical model takes into account the needle deflection; therefore, with the adequate development of the control strategy, the deflection could be minimized using predictive or adaptive controllers.